Totally Disconnected L.C. Groups: Subgroups associated with an - - PowerPoint PPT Presentation

Totally Disconnected L.C. Groups: Subgroups associated with an automorphism

George Willis The University of Newcastle February 10th − 14th 2014

Lecture 1: The scale and minimizing subgroups for an endomorphism Lecture 2: Tidy subgroups and the scale Lecture 3: Subgroups associated with an automorphism Contraction groups The structure of closed contraction groups The nub of α Lecture 4: Flat groups of automorphisms

The contraction group for α

Definition

Let α ∈ Aut(G). The contraction group for α is con(α) := {x ∈ G | αn(x) → 1 as n → ∞} . Then con(α) is an α-stable subgroup of G. Examples show that it need not be a closed subgroup.

Examples of contraction groups

Examples

• 1. F Z, where F is a finite group, with the product topology.

Let α be the shift: α(g)n = gn+1.

• 2. (Fp((t)), +), the additive group of the field of formal

Laurent series over the field of order p. Let α be multiplication by t.

• 3. Aut(Tq), the automorphism group of the regular tree with

every vertex having valency q. Let α be the inner automorphism αg, g a translation of T.

• 4. SL(n, Qp), the special linear group over the field of p-adic

numbers. Let α be conjugation by p 1

• .

Contraction groups in representation theory

Proposition (Mautner phenomenon)

Let ρ : G → L(X) be a bounded, strongly continuous representation of G on the Banach space X. Suppose, for some g ∈ G and x ∈ X, that ρ(g)x = x. Then ρ(h)x = x for every h ∈ con(g).

Non-triviality of con(α)

The following were shown by U. Baumgartner & W. in the case when G is metrizable. The metrizability condition was removed by W. Jaworski.

Theorem

Suppose that s(α−1) > 1. Then con(α) is not trivial. The converse does not hold.

Theorem

Let α ∈ Aut(G) and V ∈ B(G) be tidy for α. Then V−− = V0con(α). (1) Moreover,

• {U−− | U tidy for α} = con(α).

(2)

Normal closures

Proposition

Let α ∈ Aut(G). Then the map η : con(α) → con(α) defined by η(x) = xα(x−1) is surjective.

Proposition

Let g ∈ G. Then con(g) is contained in every (abstractly) normal subgroup of G that contains g.

The Tits core

Definition

The Tits core of the t.d.l.c. group G is G† = con(g) | g ∈ G.

Theorem (Caprace, Reid & W.)

Let D be a dense subgroup of the t.d.l.c. group G. If G† normalises D, then G† ≤ D.

Corollary (Caprace, Reid & W.)

Suppose that G belongs to S, that is, G is compactly generated and topologically simple. Then G† is either trivial or it is the smallest non-trivial normal subgroup of G.

Closed contraction groups

Theorem (Glöckner & W.)

Let G be a t.d.l.c. group and suppose that α ∈ Aut(G) is such that αn(g) → 1 as n → ∞ for every g ∈ G. Then the set tor(G)

• f torsion elements and the set div(G) of divisible groups are

α-stable closed subgroups of G and G = tor(G) × div(G). Furthermore div(G) is a direct product div(G) = Gp1 × · · · × Gpn, where each Gpj is a nilpotent pi-adic Lie group.

Closed contraction groups 2

Every group G with a contractive automorphism α has a composition series of closed α-stable subgroups where each of the composition factors is a simple contraction group in the sense that it has no closed, proper, non-trivial α-stable subgroups.

Theorem (Glöckner & W.)

Let G be a t.d.l.c. group, α ∈ Aut(G) and suppose that (G, α) is

• simple. Then G is either:
• 1. a torsion group and isomorphic to F (−N) × F N0 with F a

finite simple group and α the shift; or

• 2. torsion free and isomorphic to a p-adic vector space

with α a contractive linear transformation.

Ergodic actions by automorphisms

Conjecture (Halmos)

Let G be a l.c. group and suppose that there is α ∈ Aut(G) that acts ergodically on G. Then G is compact. Proved for G connected in the 1960’s and for G totally disconnected in the 1980’s. Short proof by Previts & Wu uses the scale.

• S. G. Dani, N. Shah & W. show that, if G has a finitely

generated abelian group of automorphisms that acts ergodically, then G is, modulo a compact normal subgroup, a direct product of vector groups over R and Qp.

The largest subgroup on which α acts ergodically

Definition

The nub of α ∈ Aut(G) is the subgroup nub(α) =

• {V | V is tidy for α} (= nub(α−1)).

The nub of α is trivial if and only if con(α) is closed.

Theorem

nub(α) is the largest closed α-stable subgroup of G on which α acts ergodically.

Theorem

The compact open subgroup V is tidy below for α ∈ Aut(G) if and only if nub(α) ≤ V.

The structure of nub(α)

(B. Kitchens & K. Schmidt.

• W. Jaworski)

Theorem

The nub of α is isomorphic to an inverse limit (nub(α), α) ∼ = lim ← −(Gi, αi), where Gi is a compact t.d. group, αi ∈ Aut(Gi) and Gi has a composition series {1} = H0 < H1 < · · · < Hr = Gi,

• f αi stable subgroups, with the composition factors Hj+1/Hj

isomorphic to FZ

j , for a finite simple group Fj and the induced

automorphism the shift.

The nub and contraction groups

Theorem

Let α ∈ Aut(G). Then nub(α) = con(α) ∩ con(α−1) and nub(α) ∩ con(α) = {g ∈ con(α) | {αn(g)}n∈Z is bounded} is dense in nub(α). Denote this set by bcon(α). The intersection bcon(α) ∩ bcon(α−1) need not be dense in nub(α) but nub(α)/bcon(α) ∩ bcon(α−1) is abelian.

References

1.

• N. Aoki, ‘Dense orbits of automorphisms and compactness of groups’, Topology Appl. 20 (1985), 1–15.

2.

• U. Baumgartner & G. Willis, ‘Contraction groups for automorphisms of totally disconnected groups’, Israel J.

Math., 142 (2004), 221–248. 3. P .-E. Caprace, C. Reid and G. Willis, ‘Limits of contraction groups and the Tits core’, arXiv:1304.6246. 4.

• S. Dani, N. Shah & G. Willis, ‘Locally compact groups with dense orbits under Zd -actions by

automorphisms’, Ergodic Theory & Dynamical Systems, 26 (2006), 1443–1465. 5.

• J. Dixon, M. Du Sautoy, A. Mann & D. Segal, Analytic pro-p groups, Cambridge Studies in Advanced

Mathematics 61. 6.

• H. Glöckner & G. Willis, ‘Classification of the simple factors appearing in composition series of totally

disconnected contraction groups’, J. Reine Angew. Math., 634 (2010), 141–169. 7. P . Halmos, Lectures on Ergodic Theory, Publ. Math. Soc. Japan, Tokyo (1956). 8.

• W. Jaworski, Contraction groups, ergodicity, and distal properties of automorphisms of compact groups,

preprint. 9.

• M. Lazard, ‘Groupes analytiques p-adiques’, Publications mathématiques de l’I.H.É.S., 26(1965), 5–219.

10.

• W. Previts & T.-S. Wu, ‘Dense orbits and compactness of groups’, Bull. Austral. Math. Soc., 68(1) (2003),

155–159. 11.

• K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, Basel, (1995).

12.

• G. Willis, The nub of an automorphism of a totally disconnected, locally compact group, Ergodic Theory &

Dynamical Systems, C.U.P . Firstview (2013) 1–30.